Introduction

Equations that contain parentheses often look more complicated than they really are.
In this article, you will learn how to:

• Use the distributive property to remove parentheses
• Simplify expressions before solving
• Solve two-step equations that arise after distributing
• Check your solutions

This article assumes you already know:

• The distributive property ($a(b+c) = ab + ac$)
• How to solve two-step equations such as $3x + 5 = 17$

Why Parentheses Appear in Equations

Parentheses usually appear when:

• A number is multiplied by a sum or difference
• A real-world situation involves repeated groups
• An expression needs to be simplified before solving

Examples:

• $3(x + 4)$
• $-2(5 - x)$
• $\frac{1}{2}(6x - 8)$

Parentheses tell you to treat the inside as a single unit.

The Distributive Property

The distributive property lets you remove parentheses by multiplying the outside number by each term inside:

• $a(b + c) = ab + ac$
• $a(b - c) = ab - ac$

Examples:

• $4(x + 3) = 4x + 12$
• $-2(x - 5) = -2x + 10$
• $\frac{1}{3}(6x + 9) = 2x + 3$

This step is essential before solving.

Solving Equations with Parentheses

To solve equations with parentheses:

1. Distribute
2. Combine like terms (if needed)
3. Solve the resulting two-step equation
4. Check your answer

Example 1

Solve: $$3(x + 4) = 27$$

• Distribute: $3x + 12 = 27$
• Subtract 12: $3x = 15$
• Divide by 3: $x = 5$

Example 2

Solve: $$-2(3x - 1) = 10$$

• Distribute: $-6x + 2 = 10$
• Subtract 2: $-6x = 8$
• Divide by $-6$: $x = -\frac{4}{3}$

Common Mistakes to Avoid

• Forgetting to distribute to every term
• Losing negative signs
• Distributing incorrectly when fractions are involved
• Forgetting to simplify before solving

Quick checks:

• Does every term inside parentheses get multiplied?
• Are signs handled correctly?
• Is the equation simplified before solving?

Exercises

1. Solve: $4(x - 3) = 20$

   Solution

   $4(x - 3) = 20$

   • $4x - 12 = 20$
   • $4x = 32$
   • $x = 8$
2. Solve: $-2(2x + 5) = 6$

   Solution

   $-2(2x + 5) = 6$

   • $-4x - 10 = 6$
   • $-4x = 16$
   • $x = -4$
3. Solve: $\frac{1}{3}(9x - 6) = 10$

   Solution

   $\frac{1}{3}(9x - 6) = 10$

   • $3x - 2 = 10$
   • $3x = 12$
   • $x = 4$
4. Solve: $5 - 3(x + 2) = -7$

   Solution

   $5 - 3(x + 2) = -7$

   • $5 - 3x - 6 = -7$
   • $-3x - 1 = -7$
   • $-3x = -6$
   • $x = 2$
5. Solve: $7(x - 4) + 2 = 16$

   Solution

   $7(x - 4) + 2 = 16$

   • $7x - 28 + 2 = 16$
   • $7x - 26 = 16$
   • $7x = 42$
   • $x = 6$
6. Solve: $-4(2 - x) = 12$

   Solution

   $-4(2 - x) = 12$

   • $-8 + 4x = 12$
   • $4x = 20$
   • $x = 5$
7. Solve: $3(x + 5) - 2x = 11$

   Solution

   $3(x + 5) - 2x = 11$

   • $3x + 15 - 2x = 11$
   • $x + 15 = 11$
   • $x = -4$
8. Solve: $\frac{1}{2}(4x + 10) - 3 = 7$

   Solution

   $\frac{1}{2}(4x + 10) - 3 = 7$

   • $2x + 5 - 3 = 7$
   • $2x + 2 = 7$
   • $2x = 5$
   • $x = \frac{5}{2}$